Abstract
Publisher Summary This chapter describes the current state of the foundations of mathematics. There is clear evidence that the way in which doubts (about a piece of mathematics) are resolved is that the doubtful notions or inferences are refined and clarified to the point where they can be taken as proofs and definitions from existing notions within some first-order theory. A foundation of mathematics is the discussion of the first order systems within which mathematics is set out. Discussion of which notions are needed for presenting a piece of mathematics would fall under foundations, as does the notion of consistency strength.
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